We show that if A is a closed linear operator defined in a Banach space X and there exist $t_{0} \geq 0$ and $M>0$ such that $\{(im)^{\alpha }\}_{|m|> t_{0}} \subset \rho (A)$ , the resolvent set of A, and $$ \bigl\Vert (im)^{\alpha }\bigl(A+(im)^{\alpha }I \bigr)^{-1} \bigr\Vert \leq M \quad \text{ for all } \vert m \vert > t_{0}, m \in \mathbb{Z}, $$ then, for each $\frac{1}{p}<\alpha \leq \frac{2}{p}$ and $1< p < 2$ , the abstract Cauchy problem with periodic boundary conditions $$ \textstyle\begin{cases} _{GL}D^{\alpha }_{t} u(t) + Au(t) = f(t), & t \in (0,2\pi ); \\ u(0)=u(2\pi ), \end{cases} $$ where $_{GL}D^{\alpha }$ denotes the Grünwald–Letnikov derivative, admits a normal 2π-periodic solution for each $f\in L^{p}_{2\pi }(\mathbb{R}, X)$ that satisfies appropriate conditions. In particular, this happens if A is a sectorial operator with spectral angle $\phi _{A} \in (0, \alpha \pi /2)$ and $\int _{0}^{2\pi } f(t)\,dt \in \operatorname{Ran}(A)$ .

中文翻译：

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分数抽象柯西问题的正态周期解

我们证明如果 A 是在 Banach 空间 X 中定义的闭线性算子，并且存在 $t_{0} \geq 0$ 和 $M>0$ 使得 $\{(im)^{\alpha }\}_ {|m|> t_{0}} \subset \rho (A)$ ，A 的解析集，$$ \bigl\Vert (im)^{\alpha }\bigl(A+(im)^{\ alpha }I \bigr)^{-1} \bigr\Vert \leq M \quad \text{ for all } \vert m \vert > t_{0}, m \in \mathbb{Z}, $$ 然后，对于每个 $\frac{1}{p}<\alpha \leq \frac{2}{p}$ 和 $1< p < 2$ ，具有周期性边界条件的抽象柯西问题 $$ \textstyle\begin{cases} _{GL}D^{\alpha }_{t} u(t) + Au(t) = f(t), & t \in (0,2\pi ); \\ u(0)=u(2\pi ), \end{cases} $$ 其中 $_{GL}D^{\alpha }$ 表示 Grünwald–Letnikov 导数，承认每个的正常 2π-周期解$f\in L^{p}_{2\pi }(\mathbb{R}, X)$ 满足适当条件。特别是，